Summary: From Grushin to Heisenberg via an
Nicola Arcozzi and Annalisa Baldi
The Grushin plane is a right quotient of the Heisenberg group.
Heisenberg geodesics' projections are solutions of an isoperimetric
problem in the Grushin plane.
It is a known fact that there is a correspondence between isoperi-
metric problems in Riemannian surfaces and sub-Riemannian geome-
tries in three-dimensional manifolds. The most significant example
is the isoperimetric problem in the plane, corresponding to the sub-
Riemannian geometry of the Heisenberg group H.
We briefly recall this connection following the exposition in [Mont].
Consider, on the Euclidean plane, the one-form = 1
2 (xdy - ydx),
which satisfies d = dx dy and which vanishes on straight lines
through the origin. By Stokes' Theorem, the signed area enclosed by
a curve is . Let c : [a, b] R2 be a curve. For each s in [a, b],
let s be the union of the curve c restricted to [a, s], of the segment